Class 8 Maths Chapter 5: Number Play Multiples, Remainders & Divisibility Shortcuts

 Class 8 Maths – Chapter 5: Number Play
(Ganita Prakash Part 1)

Introduction

This chapter explores the fascinating world of numbers — multiples, remainders, parity, and divisibility shortcuts. Students learn how to reason algebraically, identify patterns, and solve problems using logical deduction.

Basic Knowledge Required

• Addition, subtraction, multiplication, division

• Factors and multiples

• Even and odd numbers (parity)

• Remainders in division

• Algebraic expressions

Important Definitions

Multiple

A number obtained by multiplying a given number by an integer.

Factor

A number that divides another number exactly.

Parity

Whether a number is even or odd.

Divisibility Rule

A shortcut to check if a number is divisible by another without full division.

Formulas / Concepts Used

• Even ± Even = Even

• Odd ± Odd = Even

• Odd ± Even = Odd

• Sum of digits divisible by 9 → number divisible by 9

• Sum of digits divisible by 3 → number divisible by 3

• Alternating sum of digits divisible by 11 → number divisible by 11

• If a divides M and N → a divides (M+N) and (M−N)

• LCM rule: If A divisible by k and m → A divisible by LCM(k,m)

Solved Examples (Step‑by‑Step)

Example 1: Sum of four consecutive numbers = 34

Let numbers = n, n+1, n+2, n+3

Sum = 4n + 6 = 34

4n = 28 → n = 7

Numbers = 7, 8, 9, 10

Example 2: Five consecutive numbers with greatest = p

Numbers = p, p−1, p−2, p−3, p−4

Example 3: Remainder problem

Find numbers leaving remainder 3 when divided by 5.

General form = 5k + 3

Examples: 3, 8, 13, 18, 23

Example 4: Divisibility by 9

Check 427.

Sum of digits = 4+2+7 = 13 → remainder 4.

So 427 is not divisible by 9.

Example 5: Divisibility by 11

Check 462.

(4+2) − 6 = 0 → divisible by 11.

FIGURE IT OUT — COMPLETE SOLUTIONS

1. Four consecutive numbers sum = 34

Answer: 7, 8, 9, 10

2. Greatest of five consecutive numbers = p

Others: p−1, p−2, p−3, p−4

3. Always / Sometimes / Never

(i) Sum of two even numbers multiple of 3 → Sometimes

(ii) Not divisible by 18 → not divisible by 9 → Always true

(iii) Two not divisible by 6 → sum not divisible by 6 → Sometimes

(iv) Multiple of 6 + multiple of 9 → multiple of 3 → Always true

(v) Multiple of 6 + multiple of 3 → multiple of 9 → Sometimes

4. Numbers leaving remainder 2 with 3 and 4

General form: 12k + 2

Examples: 2, 14, 26, 38

5. Pebble riddle

Answer: 105 pebbles (fits conditions with 3, 2, 5, 7 grouping)

6. Tathagat’s claim

Numbers ≡ 2 (mod 6).

Sum of three such numbers ≡ 6 (mod 6) → divisible by 6.

Claim is true.

7. Remainders with 7

661 ≡ 3, 4779 ≡ 5

(i) 4779+661 ≡ 8 ≡ 1

(ii) 4779−661 ≡ 2

8. Smallest number with remainders (3 mod 4, 2 mod 3, 4 mod 5)

Answer: 59

Conclusion

Number Play builds strong foundations in multiples, remainders, parity, and divisibility. These shortcuts and algebraic reasoning techniques make problem solving efficient and fun.

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FAQs – Chapter 5: Number Play

1. What is parity?

Parity refers to whether a number is even or odd.

2. Can all numbers be written as sum of consecutive numbers?

Not all, but many can. Odd numbers = sum of two consecutive numbers.

3. What is the sum of angles in a quadrilateral?

Always 360°. (From earlier chapters, useful here.)

4. What is the shortcut for divisibility by 9?

A number is divisible by 9 if the sum of its digits is divisible by 9.

5. What is the shortcut for divisibility by 3?

A number is divisible by 3 if the sum of its digits is divisible by 3.

6. What is the shortcut for divisibility by 11?

Alternating sum of digits divisible by 11.

7. What is the general rule for divisibility by LCM?

If A divisible by k and m, then A divisible by LCM(k,m).

8. What is the remainder form for numbers ≡ 3 mod 5?

General form = 5k + 3.

9. What is the smallest number ≡ 2 mod 3, ≡ 3 mod 4, ≡ 4 mod 5?

Answer: 59.

10. What is the sum of four consecutive numbers = 34?

Numbers = 7, 8, 9, 10.

11. What is the form of five consecutive numbers with greatest p?

p, p−1, p−2, p−3, p−4.

12. Is sum of two even numbers always multiple of 3?

No, only sometimes.

13. Is Tathagat’s claim true?

Yes, sum of three numbers ≡ 2 mod 6 is divisible by 6.

14. What is divisibility rule for 2, 5, 10?

Check units digit: 0 → divisible by 10, 0/5 → divisible by 5, even digit → divisible by 2.

15. What is the shortcut for divisibility by 4 and 8?

Check last 2 digits for 4, last 3 digits for 8.